Theorem summodclem2 11737

Description: Lemma for summodc 11738. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)

Hypotheses

Ref	Expression
isummo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
isummo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
summodclem2.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))

Assertion

Ref	Expression
summodclem2	⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))

Distinct variable groups:   𝑘,𝑛,𝐴   𝑛,𝐹   𝜑,𝑘,𝑛   𝐴,𝑓,𝑗,𝑚,𝑘,𝑛   𝐵,𝑛   𝑓,𝐹,𝑘,𝑚   𝜑,𝑓,𝑚   𝑥,𝑓,𝑘,𝑚,𝑛   𝑦,𝑓,𝑚

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑗)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦,𝑓,𝑗,𝑘,𝑚)   𝐹(𝑥,𝑦,𝑗)   𝐺(𝑥,𝑦,𝑓,𝑗,𝑘,𝑚,𝑛)

Proof of Theorem summodclem2

Dummy variables 𝑎 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5583	. . . . 5 ⊢ (𝑚 = 𝑎 → (ℤ≥‘𝑚) = (ℤ≥‘𝑎))
2	1	sseq2d 3224	. . . 4 ⊢ (𝑚 = 𝑎 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐴 ⊆ (ℤ≥‘𝑎)))
3	1	raleqdv 2709	. . . 4 ⊢ (𝑚 = 𝑎 → (∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴))
4		seqeq1 10602	. . . . 5 ⊢ (𝑚 = 𝑎 → seq𝑚( + , 𝐹) = seq𝑎( + , 𝐹))
5	4	breq1d 4057	. . . 4 ⊢ (𝑚 = 𝑎 → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑎( + , 𝐹) ⇝ 𝑥))
6	2, 3, 5	3anbi123d 1325	. . 3 ⊢ (𝑚 = 𝑎 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)))
7	6	cbvrexv 2740	. 2 ⊢ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ ∃𝑎 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥))
8		simplr3 1044	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → seq𝑎( + , 𝐹) ⇝ 𝑥)
9		simplr1 1042	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ⊆ (ℤ≥‘𝑎))
10		uzssz 9675	. . . . . . . . . . . 12 ⊢ (ℤ≥‘𝑎) ⊆ ℤ
11	9, 10	sstrdi 3206	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ⊆ ℤ)
12		1zzd 9406	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 1 ∈ ℤ)
13		simprl 529	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑚 ∈ ℕ)
14	13	nnzd 9501	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑚 ∈ ℤ)
15	12, 14	fzfigd 10583	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (1...𝑚) ∈ Fin)
16		simprr 531	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
17		f1oeng 6855	. . . . . . . . . . . . . 14 ⊢ (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (1...𝑚) ≈ 𝐴)
18	15, 16, 17	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (1...𝑚) ≈ 𝐴)
19	18	ensymd 6882	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ≈ (1...𝑚))
20		enfii 6978	. . . . . . . . . . . 12 ⊢ (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
21	15, 19, 20	syl2anc 411	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ∈ Fin)
22		zfz1iso 10993	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
23	11, 21, 22	syl2anc 411	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
24		isummo.1	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
25		simplll 533	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑)
26		isummo.2	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
27	25, 26	sylan 283	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
28		eleq1w 2267	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
29	28	dcbid 840	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑘 ∈ 𝐴))
30		simpr2 1007	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴)
31	30	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ≥‘𝑎)) → ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴)
32		simpr 110	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ≥‘𝑎)) → 𝑘 ∈ (ℤ≥‘𝑎))
33	29, 31, 32	rspcdva 2883	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ≥‘𝑎)) → DECID 𝑘 ∈ 𝐴)
34		summodclem2.g	. . . . . . . . . . . . 13 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
35		eqid 2206	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑔‘𝑛) / 𝑘⦌𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑔‘𝑛) / 𝑘⦌𝐵, 0))
36		simprll 537	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
37		simpllr 534	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑎 ∈ ℤ)
38		simplr1 1042	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ≥‘𝑎))
39		simprlr 538	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
40		simprr 531	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
41	24, 27, 33, 34, 35, 36, 37, 38, 39, 40	summodclem2a 11736	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚))
42	41	expr 375	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)))
43	42	exlimdv 1843	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)))
44	23, 43	mpd 13	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚))
45		climuni 11648	. . . . . . . . 9 ⊢ ((seq𝑎( + , 𝐹) ⇝ 𝑥 ∧ seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
46	8, 44, 45	syl2anc 411	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
47	46	anassrs 400	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
48		eqeq2 2216	. . . . . . 7 ⊢ (𝑦 = (seq1( + , 𝐺)‘𝑚) → (𝑥 = 𝑦 ↔ 𝑥 = (seq1( + , 𝐺)‘𝑚)))
49	47, 48	syl5ibrcom 157	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑦 = (seq1( + , 𝐺)‘𝑚) → 𝑥 = 𝑦))
50	49	expimpd 363	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
51	50	exlimdv 1843	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
52	51	rexlimdva 2624	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
53	52	r19.29an 2649	. 2 ⊢ ((𝜑 ∧ ∃𝑎 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
54	7, 53	sylan2b 287	1 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 836   ∧ w3a 981   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  ⦋csb 3094   ⊆ wss 3167  ifcif 3572   class class class wbr 4047   ↦ cmpt 4109  –1-1-onto→wf1o 5275  ‘cfv 5276   Isom wiso 5277  (class class class)co 5951   ≈ cen 6832  Fincfn 6834  ℂcc 7930  0cc0 7932  1c1 7933   + caddc 7935   < clt 8114   ≤ cle 8115  ℕcn 9043  ℤcz 9379  ℤ_≥cuz 9655  ...cfz 10137  seqcseq 10599  ♯chash 10927   ⇝ cli 11633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-irdg 6463  df-frec 6484  df-1o 6509  df-oadd 6513  df-er 6627  df-en 6835  df-dom 6836  df-fin 6837  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-n0 9303  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  df-fz 10138  df-fzo 10272  df-seqfrec 10600  df-exp 10691  df-ihash 10928  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354  df-clim 11634

This theorem is referenced by:  summodc  11738